We study nonequilibrium steady states in the Lorentz gas of periodic s
catterers when an electric external field is applied and the particle
kinetic energy is held fixed by a ''thermostat'' constructed according
to Gauss' principle of least constraint (a model problem previously s
tudied numerically by Moran and Hoover). The resulting dynamics is rev
ersible and deterministic, but does not preserve Liouville measure. Fo
r a sufficiently small field, we prove the following results: (1) exis
tence of a unique stationary, ergodic measure obtained by forward evol
ution of initial absolutely continuous distributions, for which the Pe
sin entropy formula and Young's expression for the fractal dimension a
re valid; (2) exact identity of the steady-state thermodynamic entropy
production, the asymptotic decay of the Gibbs entropy for the time-ev
olved distribution, and minus the sum of the Lyapunov exponents; (3) a
n explicit expression for the full nonlinear current response (Kawasak
i formula); and (4) validity of linear response theory and Ohm's trans
port law, including the Einstein relation between conductivity and dif
fusion matrices. Results (2) and (4) yield also a direct relation betw
een Lyapunov exponents and zero-field transport ( = diffusion) coeffic
ients. Although we restrict ourselves here to dimension d = 2, the res
ults carry over to higher dimensions and to some other physical situat
ions: e.g. with additional external magnetic fields. The proofs use a
well-developed theory of small perturbations of hyperbolic dynamical s
ystems and the method of Markov sieves, an approximation of Markov par
titions.